Bloom’s inequality: commutators in a two-weight setting

Abstract

In 1985, Bloom characterized the boundedness of the commutator [b, H] as a map between a pair of weighted L p spaces, where both weights are in A p . The characterization is in terms of a novel BM O condition. We give a 'modern' proof of this result, in the case of p = 2. In a subsequent paper, this argument will be used to generalize Bloom's result to all Calderón-Zygmund operators and dimensions. 1 2 . 2000 Mathematics Subject Classification. Primary .

Boundedness of Commutators and H $${}^1$$ 1 -BMO Duality in the Two Matrix Weighted Setting

“…As in [9,10,13], we will prove sufficiency in Theorem 1.1 by proving two matrix weighted norm inequalities for dyadic paraproducts in terms of equivalent BMO conditions similar to the ones in [9,10] (and when p = 2 in particular prove a two matrix weighted generalization of Theorem 3.1 in [9].) Given a matrix weight W , let V I (W, p) and V ′ I (W, p) be reducing operators satisfying |I|…”

“…Moreover, it is instructive and quite interesting to compare Theorem 2.2 when p = 2 to Theorem 3.1 of [9] in the scalar setting. In particular it was shown in [9] that a scalar symbolled paraproduct π b : L 2 (u) → L 2 (w) for two scalar A 2 weights w and u if and only if…”

“…Let us briefly comment on the ideas and techniques used in this paper. Like [9,10], the ideas and techniques in this paper are "dyadic" in nature and are very different than the more classical ideas and techniques in [3]. However, since the techniques in [9,10] are obviously scalar weighted techniques, we will not draw from them in this paper, but instead heavily rely on the ideas developed in two recent preprints, the first being [13] by the author, H. K. Kwon, and Sandra Pott, and the second being [12] by the author.…”

“…Like [9,10], the ideas and techniques in this paper are "dyadic" in nature and are very different than the more classical ideas and techniques in [3]. However, since the techniques in [9,10] are obviously scalar weighted techniques, we will not draw from them in this paper, but instead heavily rely on the ideas developed in two recent preprints, the first being [13] by the author, H. K. Kwon, and Sandra Pott, and the second being [12] by the author. It should be commented, however, that we will in fact use the papers [9,10] as a sort of "guiding light" for recasting the various matrix weighted BMO conditions in [12,13] into two matrix weighted conditions.…”

“…However, since the techniques in [9,10] are obviously scalar weighted techniques, we will not draw from them in this paper, but instead heavily rely on the ideas developed in two recent preprints, the first being [13] by the author, H. K. Kwon, and Sandra Pott, and the second being [12] by the author. It should be commented, however, that we will in fact use the papers [9,10] as a sort of "guiding light" for recasting the various matrix weighted BMO conditions in [12,13] into two matrix weighted conditions. Also note that with this in mind, one can think of this paper as a kind of "two weight unification" of some of the ideas and results in [12,13].…”